spacelike, timelike and lightlike [...] What, however, is the physical intuition behind?

Using "physical" terminology means (foremost, and even exclusively) to refer to distinguishable "participants" (a.k.a. "principal identifiable points", or "material points") where

• each is (thought as being) capable, at least in principle, of determining with whom they had been coincident ("meeting at an event"), and with whom not, and in which sequence one had taken part in different concidence events, and

• each is (thought as being) capable, at least in principle, of observing and recognizing others in their distinguishable states (having taken part in particular coincidence events, having collected certain observations), thus exchanging signals between each other (i.e. especially referring to one's first observation of a given signal state).

In this terminology, any one event is characterized by who had taken part, and by which signals (of other events) the participants first observed at this coincidence occasion.

The relation between two distinct given events may accordingly be characterized as follows:

• either all participants in one event had thereby first observed the (signals of) the other event. Such pairs of events are conventionally characterized by a null interval,

• or else at least one identifiable participants having taken part in both events (or at least being thought as having taken part in both events). Considering three or more such events they are conventionally assigned suitably generalized metric relations between each other to satisfy the inverse triangle inequality,

• or neither. Metric relations assigned to three or more such events may or may not satisfy the triangle inequality.

However, as technical terms, the words "spacelike, timelike and lightlike" require some (not quite arbitrary) additional assignment of coordinates to a given set of events, in order to describe their relation as elements of a Lorentzian manifold.

spacelike, timelike and lightlike [...] What, however, is the physical intuition behind?

Using "physical" terminology means (foremost, and even exclusively) to refer to distinguishable "participants" (a.k.a. "principal identifiable points", or "material points") where

• each is (thought as being) capable, at least in principle, of determining with whom they had been coincident ("meeting at an event"), and with whom not, and in which sequence one had taken part in different concidence events, and

• each is (thought as being) capable, at least in principle, of observing and recognizing others in their distinguishable states (having taken part in particular coincidence events, having collected certain observations), thus exchanging signals between each other (i.e. especially referring to one's first observation of a given signal state).

In this terminology, any one event is characterized by who had taken part, and by which signals (of other events) the participants first observed at this coincidence occasion.

The relation between two distinct given events may accordingly be characterized as follows:

• either all participants in one event had thereby first observed the (signals of) the other event. Such pairs of events are conventionally characterized by a null interval,

• or else at least one identifiable participant having taken part in both events (or at least being thought as having taken part in both events). Considering three or more such events they are conventionally assigned suitably generalized metric relations between each other to satisfy the inverse triangle inequality,

• or neither. Metric relations assigned to three or more such events may or may not satisfy the triangle inequality.

However, as technical terms, the words "spacelike, timelike and lightlike" require some (not quite arbitrary) additional assignment of coordinates to a given set of events, in order to describe their relation as elements of a Lorentzian manifold.